This would depend on both your definition of success and system. People also look for alternatives if they are not satisfied with the options presented to them. In a market economy, this means changing the parties (in the legal since) you do business with. However, when options are restricted, people will find alternatives. For instance, many Canadians, where private healthcare is illegal, seek private treatment in the United States when they are sufficiently dissatisfied with the options provided for them. Furthermore, private healthcare exists in the UK; patients are more likely to use private services here when healthcare is rationed. In fact, I would argue the current US system is a good example of how private healthcare would work; the business of healthcare will adapt to the incentives. The questions is who sets them: the market or politics.
RE: Week 10 -- Question about private healthcare in a universal healthcare system